Instruction for Mathematical Knowledge for Teachers of ElementaryMiddle Grades

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Instruction for Mathematical Knowledge for
Teachers of Elementary/Middle Grades
Melissa Hedges Hank Kepner Gary Luck Kevin
McLeod Lee Ann Pruske UW-Milwaukee
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UWM Foundational Courses for 1-8 Education Majors

• MATH 175 Mathematical Explorations for
  Elementary Teachers, I
• MATH 175 Mathematical Explorations for
  Elementary Teachers, II
• CURRINS 331 Teaching of Mathematics Grades 1-6
• CURRINS 332 Teaching of Mathematics Middle
  School

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UWM Math Focus Courses for 1-8 Education Majors

• MATH 275 Problem-Solving and Critical Thinking
• MATH 276 Algebraic Structures
• MATH 277 Geometry
• MATH 278 Discrete Probability and Statistics
• (Over 40 of UWM 1-8 Education Majors choose a
  mathematics focus area)

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Course Design Team Model

• Mathematics faculty member ensures rigorous
  content
• Mathematics Education faculty member ensures
  strong pedagogy, and alignment with standards
• Teacher-in-Residence provides connection to
  classroom practice

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Topics Covered in MATH 175

• Problem-solving
• Number systems
• Fractions
• Decimals and percent
• Addition and Subtraction (meaning, and
  properties)
• Multiplication and Division (meaning, and
  properties)

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Geometry Topics Covered in MATH 176

• Visualization (solids nets)
• Angles, circles, spheres, triangles, polygons
• Constructions (patty paper Cabri on TI-84)
• Congruence and similarity
• Transformations (flips, slides, turns patty
  paper Cabri)
• Measurement
• Area (derivation of formulas Pythagoras)

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Probability and Statistics Topics Covered in MATH
176

• Plots (line plots histograms stem-and-leaf
  plots box-and-whisker plots)
• Mean, median, mode standard deviation
• Inference
• Displaying outcomes (arrays trees sample
  spaces)
• Probability (experimental theoretical)
• Simulation (ProbSim applications on TI-84)
• Games (fair/unfair relationship to probability)
• Counting principles
• Expected value

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Mathematical Topics Covered in CURRINS 331/2

• CURRINS 331 Number and operations (number
  development, place value, CGI, operation
  concepts) Computing devices Algebraic reasoning
  (patterns, computational/relational thinking)
• CURRINS 332 Geometry Algebra (linear
  equations) Probability Fractions, decimals and
  percents

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1-8 Teacher Content

• If we spin the spinner shown below many, many
  times, how many points would we average per spin?
• What is your guess? _____

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8
1
10
1-8 Teacher Content

• Lets begin with an easier example Perhaps it
  will lead us to an answer to the previous
  question.
• If we spin the spinner shown below many, many
  times, how many points would we average per spin?
• What is your guess? ____ Why?

3
8
1
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1-8 Teacher Content

• What are the similarities and the differences in
  these 2 problems?
• Similarities Differences

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1-8 Teacher Content

• Suppose we would do a simulation of this problem.
• Draw a frequency histogram that you might expect
  to get from spinning the spinner 100 times
• Why did you construct the histogram as you did?

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1-8 Teacher Content

• One such simulation produced the following
  results

50
10
1
3
8
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1-8 Teacher Content

• Now, calculate the experimental average points
  per spin from the data collected

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1-8 Teacher Content

• Now, lets calculate the theoretical number of
  points per spin (or the Expected Value)

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1-8 Teacher Content

• To calculate the Expected Value, we might
  consider the following

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1-8 Teacher Content

• What is the relationship between the 2 previous
  examples?
• 1 x 1 3 x 1 8 x 2 1 3 16 20 5
• 4 4
  4
• 1 x ¼ 3 x ¼ 8 x ½ ¼ ¾ 4 5
• Are these procedures equivalent?
• Compare this to the calculation of the
  experimental average.
• 1 x 22 2 x 26 8 x 52 490 4.9
• 100 100

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1-8 Teacher Content

• What topics in mathematics for K-8 teachers did
  we address in this activity?

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Changes to MATH 175/6

• Stabilization of instruction (hiring of Luck,
  Mandell)
• Improved instruction modeling pedagogy
• More hands-on activities (e.g. patty paper),
  resulting in greater familiarity in CURRINS 331/2

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Changes to CURRINS 331/2

• Prerequisite of C or better in MATH 176
• Stronger connections to the mathematics taught in
  MATH 175/6, including
• Greater emphasis on mathematical concepts
  (distributive law, not FOIL expressions vs.
  equations opposite vs. inverse)
• Greater emphasis on correct notation (use of
  sign to indicate balance)
• Use of definitions from MATH 175/6 textbook